A226731 a(n) = (2n - 1)!/(2n).

20, 630, 36288, 3326400, 444787200, 81729648000, 19760412672000, 6082255020441600, 2322315553259520000, 1077167364120207360000, 596585001666576384000000, 388888194657798291456000000

Offset: 3

Author

Wesley Ivan Hurt, Jun 15 2013

Keywords

• nonn
• easy

Comments

For n < 3, the formula does not produce an integer.

The ratio of the product of the partition parts of 2n into exactly two parts to the sum of the partition parts of 2n into exactly two parts. For example, a(3) = 20, and 2*3 = 6 has 3 partitions into exactly two parts: (5,1), (4,2), (3,3). Forming the ratio of product to sum (of parts), we have (5*1*4*2*3*3)/(5+1+4+2+3+3) = 360/18 = 20. - Wesley Ivan Hurt, Jun 24 2013

Examples

			a(3) = (2*3 - 1)!/(2*3) = 5!/6 = 120/6 = 20.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 3..225
• Index entries for sequences related to partitions.

Crossrefs

Cf. A001105, A002674, A005843, A009445, A093353, A143623, A211374.

Programs

Maple

seq((2*k-1)!/(2*k),k=1..20); # Wesley Ivan Hurt, Jun 24 2013

Mathematica

Table[(2n-1)!/(2n),{n,3,20}] (* Harvey P. Dale, Jun 19 2013 *)

PARI

a(n) = (2*n-1)!/(2*n); \\ Michel Marcus, Nov 01 2016

Formula

a(n) = A009445(n-1)/A005843(n) = A002674(n)/A001105(n). - Wesley Ivan Hurt, Jun 24 2013

a(n) ~ sqrt(Pi)*2^(2*n-1)*n^(2*n-3/2)/exp(2*n). - Ilya Gutkovskiy, Nov 01 2016

From Amiram Eldar, Mar 10 2022: (Start)

Sum_{n>=3} 1/a(n) = e - 8/3.

Sum_{n>=3} (-1)^(n+1)/a(n) = cos(1) + sin(1) - 4/3. (End)